Section 59 Normal Distribution: Exercise

59.1 Example 1

  • Generate 30 samples from a Normal distribution with mean 0 and sd 1.

    • Plot these values.

    • Do they look as if they come from a normal distribution?

    • What is their mean and sd? What values should the mean and sd have?

[1] -0.296425
[1] 0.9029831


59.2 Example 2

  • Generate 1000 samples from a \(N(\mu=0, \sigma=1)\) distribution.

    • Does this look more like a Normal than the previous example, based on 30 values?

    • Are the sample mean and sd closer to the true values than the previous example?

    • Would you expect them to be closer?

[1] -0.0265972
[1] 0.9973377


59.3 Example 3

  • Generate 20 values from a \(N(\mu=15, \sigma=2)\) and 20 values from a\(N(\mu=16, \sigma=2)\).

    • Compare the two data sets.

    • Can you tell they come from different distributions?


59.4 Example 4

  • For a variable with a \(N(\mu=0, \sigma=1)\) distribution:

    • What is the probability a value will be greater than zero?

    • What is the probability it will be less than 1.96?

    • What is the probability it will lie in the range (-1.96,1.96).

    • If instead the variable has a \(N(\mu=10, \sigma=52)\) distribution, what is the probability a single sample will be in the range (10-1.965, 10+1.965)?


59.5 Example 5

  • Generally it can be shown that \(Prob(\mu-1.96 \sigma < X < \mu+1.96 \sigma)\) is 0.95 where \(X \sim N(mean=\mu, sd=\sigma)\).

    • Confirm this with your own choice of mean value and sd value.


59.6 Example 6

  • If the heights of adult persons in the UK are approximately normally distributed with mean=1.62m and sd=0.06m, what proportion of the population is:

    • taller than 1.7m?

    • smaller than 1.5m?

    • In a random sample of 100 adult persons how many would you expect to be taller than (a) 1.62m (b) 1.7m?